Climb To Prime

John H. Conway’s proposed a list of math problems for which he is offering $1000 for a solution.   One of those is his “Climb to a Prime” conjecture, which is stated as:

Problem 5. Climb to a Prime:

Let n be a positive integer. Write the prime factorization in the usual way, e.g. 60 = 2² · 3 · 5, in which the primes are written in increasing order, and exponents of 1 are omitted. Then bring exponents down to the line and omit all multiplication signs, obtaining a number f(n). Now repeat.

So, for example, f(60) = f(2² · 3 · 5) = 2235. Next, because 2235 = 3 · 5 · 149, it maps, under f, to 35149, and since 35149 is prime, it maps to itself. Thus 60 → 2235 → 35149 → 35149 → ..., so we have climbed to a prime, and we stop there forever. The conjecture, in which I seem to be the only believer, is that every number eventually climbs to a prime. The number 20 has not been verified to do so. Observe that 20 → 225 → 3252 → 223271 → ... , eventually getting to more than one hundred digits without yet reaching a prime!

James Davis took up the challenge and found a counterexample: 13532385396179 = 13 ⋅ 53² ⋅ 3853
This prime number fails the conjecture because its factorization maps back to itself!

You can read a few more details here:

http://www.popularmechanics.com/science/math/news/a26815/why-13532385396179-is-a-magic-number/